Corporate Liquidity Management under Moral Hazard

Transcription

1 Corporate Liquidity Management under Moral Hazard Barney Hartman-Glaser Simon Mayer Konstantin Milbradt March 15, 219 Abstract We present a model of liquidity management and financing decisions under moral hazard in which a firm accumulates cash to forestall liquidity default. When the cash balance is high, a tension arises between accumulating more cash to reduce the probability of default and providing incentives for the manager. When the cash balance is low, the firm hedges against liquidity default by transferring cash flow risk to the manager via high powered incentives. Under mild moral hazard, firms with more volatile cash flows tend to transfer less risk to the manager and hold more cash. In contrast, under severe moral hazard, an increase in cash-flow volatility exacerbates agency cost, thereby reducing firm value, overall hedging and in particular precautionary cash-holdings. Agency conflicts lead to endogenous, state-dependent refinancing costs related to the severity of the moral hazard problem. Financially constrained firms pay low wages and instead promise the manager large rewards in case of successful refinancing. We thank Simon Board, Sebastian Gryglewicz, Julien Hugonnier and Erwan Morellec for useful comments. We also thank seminar audiences at University of Houston, EPFL, and Kellogg Northwestern. A previous version of this paper was circulated under the title Cash and Dynamic Agency. UCLA Anderson. bhglaser@anderson.ucla.edu Erasmus University Rotterdam. mayer@ese.eur.nl. Northwestern University and NBER. milbradt@northwestern.edu 1

2 1 Introduction A firm s current owners often have limited wealth and liability, and must therefore raise financing in illiquid markets to forestall running out of cash. Consequently, firms often accumulate internal cash balances to avert such a liquidity default. At the same time, accumulating a large cash balance can cause an agency problem to occur, because such balances create a larger pool from which a firm s manager can divert cash. In the standard principal-agent model in corporate finance, for example, DeMarzo and Sannikov (26), a firm s owners have deep pockets and can costlessly transfer cash into the firm at any moment to cover negative cash flow shocks. The presence of liquidity management and default in such a model then purely pertains to providing incentives to the firm s manager and does not directly speak to the accumulation of cash balances. We introduce a model in which a firm s shareholders face a trade-off between accumulating cash to prevent liquidity default and optimally providing incentives to the firm s risk-averse manager. The firm s shareholders have limited liability, cannot transfer cash into the firm after inception, and have only occasional refinancing opportunities. As a consequence, they hedge against liquidity based default by optimally managing internal cash balances. In the model, the firm requires a manager to operate. This manager can inefficiently divert from both the flow and stock of cash within the firm and therefore requires incentives. The manager has constant absolute risk averse (CARA) preferences, while the shareholders are risk neutral. Nevertheless, due to the potential for liquidity default, the shareholders are effectively risk averse over the cash stock of the firm. As such, the contracting problem between the shareholders and the manager features two forces that shape the sensitivity of the manager s pay to the performance of the firm. When the firm is far from liquidity default, the manager is more risk-averse than the shareholders, and incentive provision determines the manager s optimal exposure to cash flow shocks. When the firm is close to default, the shareholders are effectively more risk averse than the manager, and the optimal contract will give the manager high-powered incentives, that is incentives above what is required to prevent cash diversion. These high-powered incentives essentially hedge the risk of liquidity default. Our assumption that investors cannot costlessly transfer cash into the firm introduces a novel restriction on the promise-keeping constraint in the standard dynamic principal-agent model (for example, DeMarzo and Sannikov (26)). Specifically, only cash within the firm and incentive compatible promises of raising cash given the opportunity can be used to fulfill the promised value 2

3 to the manager. Thus, the firm s cash balance is a commitment device that serves as collateral for the promise of future payments to the manager. In the extreme case where raising additional funds is impossible, only promises that are sufficiently collateralized by cash fulfill the promise-keeping constraint. Under the optimal contract, negative cash-flow shocks not only reduce the firm s cash position but also lower the present value of compensation the firm owes to the manager. While the manager requires some minimum level of incentives to avert cash flow diversion, the firm may hedge through labor contracts and transfer more than this minimum level of risk by providing strong incentives. Such risk-sharing or hedging demand by the firm dominates the agency problem for low cash balances. Risk-sharing is not costless; however, as increasing the variability of the manager s pay increases risk-premium the manager requires to bear such risk. The agency problem dominates hedging needs for high cash balances, leading to labor contracts that have the minimum cash-flow sensitivity to keep the agent from stealing out of the cash-flows. Therefore our first key finding is that the optimal contract provides weaker incentives when the firm holds more cash and in particular incentives decrease after positive cash-flow realizations, put differently, we find that firms with low cash-holdings provide more equity-like compensation. In addition to hedging through labor contracts, the firm can hedge liquidity risks by delaying dividend payouts and therefore accumulating more cash. Under the optimal contract, the optimal payout policy calls for a dividend whenever the firm s cash balance exceeds a threshold which we call the dividend payout boundary. Our second key finding is that the optimal dividend payout boundary decreases in the severity of the moral hazard problem. In particular, the manager s ability to divert from the firm s cash balance means that some of her compensation must be deferred, which leads to an endogenous carrying cost of cash via the risk premium that the manager applies to deferred compensation. When the moral hazard problem is more severe, that is, when the manager can divert cash with greater efficiency, the carrying cost of cash increases and the optimal dividend payout boundary decreases. Our third key finding is that under moderate moral hazard firms facing high cash-flow uncertainty do not pass on this uncertainty to management via employment contracts, but instead hedge liquidity risk by holding more cash. 1 In contrast, firms with low cash-flow uncertainty hedge more via labor contracts and provide stronger incentives to management. When moral hazard is sufficiently severe, target cash-holdings are non-monotonic in cash-flow volatility. This result arises 1 This is generally consistent with the findings of Bates et al. (29). 3

4 because an increase in cash-flow volatility also increases the cost of incentive provision, thereby decreasing firm value and reducing the overall hedging demand. Our model questions the widely held view that firms facing more severe agency conflicts should provide stronger managerial incentives. In particular, our fourth key finding is that the relationship between incentive pay and the level of moral hazard is state dependent. When the firm has a large cash balance, the strength of incentives is increasing in the severity of the moral hazard problem. This relationship reverses for firms with low cash holdings. Because more severe agency conflicts decrease the value of the firm as a going concern, liquidation becomes (relatively) less costly, decreasing a firm s hedging demand decreases to liquidity default. Consequently, the firm transfers less risk to the manager when its cash-balance is low, and the moral hazard problem is severe. Refinancing in the presence of agency conflicts imposes an endogenously-derived flotation cost to raising funds in the absence of physical refinancing costs. In our model, the firm s ability to refinance is constrained by search frictions in capital markets, as in, for example, Hugonnier et al. (214), which lead to uncertain refinancing opportunities. Under the assumption that the firm can commit to a refinancing policy ex-ante, we find that the implied refinancing costs are state-dependent, i.e., they depend on the current cash level of the firm. Ignoring for expositional purposes possible second-order effects of different payout boundaries, our fifth key finding is that a firm, depending on its cash-holdings, either refinances to below the first best, or refinances to the first best but raises more money than necessary to pay the manager a lump-sum wage payment in excess of what incentive constraints would imply. In other words, the presence of agency always distorts the decision to raise cash away from the first-best. The key to understanding latter effect is that large promises conditional on a state in which there is unlimited access to new cash lower the required wages in states in which cash is tight without violating promise keeping, thereby lowering the likelihood of liquidity default. Furthermore, in contrast to Hugonnier et al. (214), better refinancing opportunities do not reduce the firm s hedging of liquidity risk. On the one hand, increasing the firm s access to refinancing leads it to accumulate and raise less cash. On the other hand, it leads to increased hedging of liquidity risk through managerial incentive-pay in low cash states. Next, we find that when moral hazard is more severe, incentive compatibility demands highpowered incentives on average. Under these circumstances, employment contracts then absorb a large part of the liquidity risk, resulting in outside equity becoming less volatile on average. We also demonstrate that a firm s stock return volatility need not be decreasing in the firm s liquidity and 4

5 can follow a hump-shaped pattern since a financially constrained firm hedges cash-flow risk through labor contracts to a greater extent, which in turn reduces stock return volatility. Depending on how much risk the firm transfers to the manager, we get a different relationship between liquidity and volatility of stock returns. These model predictions are novel and contrast with the findings of related models of cash-management such as that of Décamps et al. (211)), who find the relationship between cash and equity return volatility to be unambiguously monotonic. Finally, the technique we use to solve our model also represents a methodological contribution. Dynamic agency problems usually introduce the manager s promised future payments as a state variable to track the agency problem. At the same time, liquidity management problems use the firm s stock of cash as a state variable to track the liquidity of the firm. Our problem thus would appear to have two state variables. While dynamic stochastic optimization problems with more than two state variables are usually hard to solve, we show how a small expansion of the allowed wage space allows for the model to collapse to a one-dimensional optimization while maintaining the liquidity-agency trade-off. The key observation is that allowing the manager to receive small negative wages, in conjunction with allowing the manager to have a savings contract that is not identically zero along the equilibrium path, relaxes the shareholders problem. Shareholders prefer to manage liquidity, in the absence of refinancing, using costly small negative wages over holding cash-buffers in excess of the incentive constraints. 2 Cash net promised risk-adjusted future wage payments readily measure the firm s financial soundness and its distance to liquidity default 2 Model Setup Cash-Flow & Earnings. We consider a 1% equity financed firm, owned by a mass of shareholders, who we also collectively refer to as the principal. To operate the business and produce cashflows from assets, the firm has to hire a manager (agent, she). Up to firm liquidation/termination at time τ, assets in place produce a cash-flow X that follows a controlled Arithmetic Brownian Motion with drift µ and volatility σ: dx t = µdt + σdz t db t, 2 Importantly, this dimensionality reduction goes beyond the absence of wealth effects, as studied by related papers considering a CARA-manager endowed with a savings technology (compare e.g. He (211), He et al. (217) or Gryglewicz et al. (217)), which usually focus without loss of generality on zero-savings contracts. 5

6 where Z is a standard Brownian Motion on the complete probability space (Ω, F, P) with filtration F = {F t : t }. Cash-flow X is fully observable, but the agent can alter its realization through her hidden action b, where db t >, as in DeMarzo and Sannikov (26), corresponds to the diversion of funds. In our model, dx t > represents operating profits and dx t < operating cost or losses. 3 Cash Holdings. The firm is liquidated when it is not able or willing to cover its operating cost, in which case shareholders recover a liquidation value L with µ r+δ L.4 Thus, liquidation entails deadweight costs, and the firm optimally retains earnings in the form of cash-holdings M to avert liquidation. Cash-holdings M are observable to both parties. Liquidation occurs when the firm runs out of cash, i.e., at time τ = inf{t : M t = }. At this point, we would like to stress that the firm could also ask the agent who is able to maintain a savings account to cover operating losses dx t < when M t =, but optimally does not do so and indeed prefers to default at time τ. In the baseline version of our model, we assume that no refinancing is possible. Later, we introduce refinancing opportunities at Poisson times and discuss the impact of agency frictions on optimal refinancing policies. Next, we assume that the firm is subject to an exogenous shock that wipes out its entire cashbalance M and trigger immediate liquidation. 5 with intensity δ. The shock arrives according to a Poisson process This assumption is needed to ensure that the model is well-behaved, in that dividend payments are not indefinitely delayed. Without loss of generality, N is observable to both parties. 6 One example of such a shock can be a large lawsuit for example, Purdue Pharma (the maker of OxyCotin) recently prepare to declare bankruptcy in response to a number of lawsuits related to the Opioid crisis. The cash-stock inside the firm grows through earned interest on the balance at the market rate r > and is directly affected by cash-flow from assets, dividend payouts to shareholders ddiv t, wage payments to the manager dw t, managerial cash-diversion db t, and catastrophic shock dn t = 1: dm t = rm t dt ddiv t dw t + µdt + σdz t M t dn t db t. (1) 3 Equivalently, we could assume that cash-flow X is only observable to the agent, who then reports cash-flow ˆX and keeps the difference X ˆX for her own use. 4 We additionally impose a lower limit in the { appendix } that ensures that liquidation is better than having the agent run the firm, which is given by L max µ ρrσ 2 /2, r+δ 5 The exact nature of the shock is not relevant as long as it triggers default with positive probability and some exogenous loss of cash. Instead of a fixed loss, we could equally assume that the shock size is exponentially distributed like in Hugonnier and Morellec (217), or impose an internal cost of carrying cash. 6 If N were only observable to the agent, the model and its solution would be entirely the same. Equivalently, we could model the disastrous shock also as cash-flow shock: dx t = µdt + σdz t db t M t dn t. 6

7 Here, M t = lim s t M t denotes the left limit of cash M t. More intuitively, M t represents cashholdings just before the catastrophic event (a jump) dn t {, 1} realizes. Moral Hazard. We assume that the manager can secretly divert cash for her own use by the following actions: First, because firm performance is noisy, i.e., σ >, the manager can secretly steal some infinitesimal amount db t > with db t o p (dt) from the firm s cash-flow dx t. By doing so, she appropriates fraction λ 1 per dollar diverted, so stealing is assumed inefficient. Conversely, the agent can also put her own money into the firm and boost cash-flow through db t <. This transfer is not subject to efficiency losses. As long as db t is infinitesimal and smooth that is, db t = ˆb t dt for some process ˆb the principal cannot detect the agent s hidden action and attributes the loss ˆbt dt due to the agent s cash-flow diversion mistakenly to a lower cash-flow shock dz t. 7 Second, the manager can divert a lumpy amount of cash smaller or equal than M t from the firm s cash-balance and in particular abscond with the entire cash-balance M t, in which case M t jumps down. Importantly, this cash diversion is immediately detected by the principal, because absent moral hazard cash-flow dx t is not subject to jump shocks and therefore continuous, and the only exogenous jump in the model, dn t, is publicly observable. We assume that the agent s benefit from stealing from the cash-stock is a fraction κ per dollar diverted. Throughout the paper, we denote the amount of cash stolen by the manager up to time t by b t and the amount received by B t, where B t does not necessarily equal b t, as diversion is subject to efficiency losses. 8 Preferences. Shareholders are risk-neutral, have zero private wealth and are protected by limited liability/commitment. That is, dividend payouts must be non-negative, that is, ddiv t for all t. 9 Shareholders discount at market rate r and maximize total firm value, which is given by discounted cumulative dividend payouts. Because shareholders cannot fully commit, they could at any time potentially pay out all cash M t, liquidate the firm and renege on the manager s promised payments. In case shareholders try to do so, we assume that the firm s cash-stock is 7 Any stochastic, i.e., non-smooth, stealing would immediately be revealed by the quadratic variation of the process, and thus is not used by the agent. 8 Formally, write db t = ˆb tdt + db 1 t, where the process ˆb is absolutely continuous. Then, db t = max{, ˆb t}λdt + min{, ˆb t}dt + κdb 1 t. 9 The zero private wealth assumption is simply to keep the shareholders from injecting cash into the firm to keep it alive. Dispersed shareholders with positive wealth in the absence of coordination would also result in the absence of cash injections due to a free-rider problem. 7

8 divided between shareholders and manager according to the Nash-Bargaining protocol, where the shareholders possess bargaining weight θ >. Likewise, shareholders cannot commit to any wage payments dw t > after liquidation for t > τ and optimally do not pay any wages after liquidation. The manager discounts at market rate r and is risk-averse with CARA-utility u(c t ) = 1 ρ exp( ρc t), where ρ > is the coefficient of absolute risk-aversion and c t is instantaneous consumption. The manager cannot fully commit and may decide to leave the firm and abscond with the entire cashbalance or any other amount, whenever she is better off from doing so. In addition, we assume that the manager can maintain hidden savings S, so that her consumption c is not observable to the principal. For tractability, we assume that the manager can borrow, implying that S t need not be positive. Savings S then earn/pay interest at rate r and are subject to changes induced by wage payments dw t, diverted cash db t, and consumption c t : ds t = rs t dt + db t + dw t c t dt (2) Endowing the agent with the possibility to accumulate savings is needed to ensure consumption smoothing beyond any liquidation event. The manager maximizes expected, discounted utility and possesses an outside option u. We normalize initial savings S = and the agent s outside option in certainty equivalence terms (properly defined in the next subsection) W =. We will make the following assumption on the wage process dw: Assumption 1. We assume that cumulative wages must satisfy lim ε w t+ε w t. That is, wages have to be either continuous or exhibit upward jumps (lumpy payments to the manager), but cannot exhibit downward jumps (lumpy cash infusions from the manager). Note that this assumption does not preclude negative flow wages. We discuss the above and alternative assumptions in more detail in Sections and The Contracting Problem. At inception t =, the manager is offered a contract C = (ĉ, w, ˆb) by the shareholders, who also decide on optimal cash-holdings and the payout process Div. The contract C specifies the manager s recommended consumption ĉ, wage payments w and diversion ˆb. We call C incentive compatible if c t = ĉ t and ˆb t = b t =, in that the manager does not (inefficiently) 8

9 steal from the firm s cash-stock, and feasible if the principal can fully commit to it. Throughout the paper, we focus on incentive compatible and feasible contracts and denote the set of these contracts by C. The agent solves for some initial savings S [ ] U = max c,b e rt u(c t )dt s.t. (2) (3) while the shareholders objective is it to maximize firm value: V = [ ] max Div,C C e rt ddiv t + e rτ L, (4) s.t. U u, ddiv t, M t for all t and (1). (5) To ensure the problem is well-behaved, we impose that the agent s savings S must satisfy the transversality condition, sometimes referred to as the No-Ponzi condition: lim t e rt S t almost surely wrt. P (6) and certain other regularity conditions, which are collectively gathered in Appendix A. If ever S τ <, the transversality condition requires negative consumption to make up the savings shortfall. 3 Model Solution In this section, we solve the model and derive the firm s optimal payout and executive compensation policy. First, we analyze the manager s problem and characterize conditions for the contract to be incentive compatible. In particular, we introduce the certainty equivalent W t. Second, we focus on the principal s problem and show the restriction on the state- and strategy-space the principal faces. In particular, due to CARA, the principal faces a 2-dimensional dynamic optimization problem characterized by a PDE. Third, we show how under Assumption 1 on wages the model collapses to a 1-dimensional dynamic optimization problem characterized by an ODE while maintaining a liquidity-default trade-off. 9

10 3.1 The manager s problem The Continuation Value As is standard in the dynamic agency literature, let us define for any incentive compatible contract C the agent s continuation value at time t U t := E t [ ] e r(s t) u(c s )ds (7) and denote the agent s savings by S t. dynamics of U follow By the martingale representation theorem, we get that du t = ru t dt u(c t )dt + β t ( ρru t ) (dx t µdt) }{{} α t ( ρru t )(dn t δdt) (8) =σdz t for some (conveniently scaled) loadings α, β defined by the contract. Here, α captures the agent s exposure to disaster risk dn t and β the agent s exposure to cash-flow shocks dz t. First, note that in order to ensure that the agent does not deviate from the recommended consumption path, the optimal contract has to respect the agent s Euler equation, in that marginal utility has to follow a martingale. Next, as shown in the appendix, the first order condition with respect to consumption with the possibility of a savings account implies that u (c t ) = ρru t >. This in turn implies that U t is a martingale. Further, let us define the certainty equivalent W t as the amount of wealth needed that would result in utility U t if the agent only consumed interest rw t, i.e., u (rw t ) = ρru t W (U) := ln( ρru). (9) ρr Here, W t is the agent s continuation value in monetary terms while U t is the agent s continuation value in utility terms. By Ito s Lemma, we obtain dw t = ρr 2 (β tσ) 2 }{{} BM Risk-Premium ( + δ α t ln(1 + ρrα t) dt + β t (dx t µdt) ) } ρr {{ } Poisson Risk-Premium> dt ln(1 + ρrα t) (dn t δdt). (1) ρr Because her compensation package is exposed to cash-flow shocks dx t and productivity shocks 1

11 dn t, the agent demands a risk premium, so that W t has a positive drift. In other words, as U t is a martingale, W t = W (U t ) has a positive drift due to the convexity of W (U) and Jensen s inequality. Essentially, (8) or equivalently (1) constitutes the so-called promise-keeping constraint. That is, shareholders promise the agent s continuation value W (resp. U) evolves according to (1) (resp. (8)) Cash & Cash-Flow Diversion In this section, we analyze the incentives the optimal contract has to provide to the manager in order to preclude diversion of cash. In principle, the agent can pursue two actions. First, she can steal some infinitesimal amount ε > of cash. When this amount is sufficiently small (on the order of dt), the principal mistakenly attributes the losses to an adverse cash-flow shock dz t < and can accordingly not detect this misbehavior. We refer to this action as cash-flow diversion. Diverting and consuming amount ε > from cash-flow increases flow utility by u (c t )λε while dx t falls by ε, so that on average the agent s continuation value is reduced by (recall our scaling of the loadings in the martingale representation) β t ( ρru t )ε = β t u (c t )ε. Thus, stealing ε dollars is not optimal if β t ( ρru t ) λu (c t ) β t λ. (11) Therefore, the principal has to provide a minimum performance-pay in order to rule out the agent diverts from cash-flow. In principle, the manager can also boost cash-flow dx t through putting in additional cash from her savings account. To prevent a violation of promise keeping, any IC contract needs to have β t 1. As we shall see, the constraint β t 1 never binds and does not affect the principal s maximization, in that the manager is optimally provided incentives β t < 1 for all t. Second, the agent can divert any larger amount cash < db t M t, in which case the firm s cash-balance jumps down by db t >. We refer to this action as cash-stock diversion, because cash-flow evolves continuously and is not subject to large shocks. Thus, absent any large shocks dn t = 1, the principal immediately observes the agent s misbehavior and can accordingly punish her. It is important here to point out that the principal does not have access to the agent s savings account, so that any punishment has to arise from decreasing future wages. In order to have some leeway to punish the agent, the principal must therefore defer compensation. Hence, whenever the firm holds a positive amount of cash M t >, i.e., for t < τ, the optimal 11

12 contract must provide incentives by means of deferred compensation, in order to preclude that the agent steals any amount from the cash-stock. Deferred compensation is represented by Y t := W t S t >, so that the agent s promised compensation exceeds her savings. We interpret Y t as the risk-adjusted value of future wages. 1 To see why Y t > discourages cash diversion, imagine that the agent considers just before time t, i.e., at t, to abscond with the entire cash-stock M t. Doing so, she receives κm t dollars, the firm is liquidated and the employment contract is terminated. Hence, after stealing, the agent does not receive any future wages, but possesses the sum of her private savings and the diverted cash S t + κm t. The agent refrains from stealing if the value from staying with the firm is higher than the value from stealing and leaving, that is if: 11 W t = S t + Y t S t + κm t Y t κm t ϕ t := Y t M t κ. (12) By (12), high cash-holdings M t within the firm exacerbate agency issues and tighten the IC constraint, which makes higher powered incentives by means of deferred payments Y t necessary. While deferring compensation by means of Y t > is necessary to align the manager s incentives, it comes at a cost. This is because during any time interval [t, t + dt] the firm might be hit by a disastrous shock, which fully exhausts the available cash-stock. In this case, the firm is liquidated and due to the shareholders limited liability the manager looses the previously promised amount Y t. By definition, at time of termination τ, the manager s certainty equivalent W τ must equal her savings S τ, i.e., Y τ =. Hence, upon a shock dn t = 1, it follows that the manager s continuation value jumps down immediately by amount Y t, in that dw t = Y t dn t. Matching coefficients in (1), this pins down the manager s exposure to disaster risk in terms of U t : α t = A(Y t ) := exp(ρry t ) 1 ρr. (13) Hence, deferring compensation exposes the manager to Poisson shocks, for which she requires a risk-premium to be paid by the firm. Consequently, increasing Y t is costly for shareholders as A( ) is increasing and convex in its argument. [ 1 It is straightforward to show Y t = E t e r(s t)( dw t s ζ )] ( ) sds where ζ t := ρr 2 (βtσ)2 + δ α t ln(1+ρrα t) is ρr the agent s required risk premium. 11 In case the agent were able to enjoy an additional outside option O in monetary terms after leaving the firm, e.g., through finding a job at another firm or through extracting some of the liquidation value of the assets, the constraint (12) would change to Y t κm t + O. Throughout our analysis, we consider without loss O = and we normalize the agent s outside option to zero. 12

13 Because higher cash-holdings M t require by (12) more deferred compensation Y t and therefore a higher risk-compensation δa(y t ) and flow wage for the manager, we obtain endogenous carrycost for internal cash-holdings. To conclude this part, we summarize our findings in the following proposition. Proposition 1. Let C solve (5). Then, the following holds true: i) The agent s continuation value U, defined in (7) solves the SDE (8) for some F-progressive processes (α, β) and W solves the SDE (1). ii) Given a process Y the process α satisfies (13). iii) The process β satisfies β t [λ, 1] for all t and the process α is given through (13). 3.2 The shareholders problem First reduction of the State Space The problem of shareholders generally depends on three states. The agent s continuation value U t or equivalently W t, the agent s savings S t and the firm s cash-holdings M t, so that firm value at time t or equivalently the shareholders continuation value is given by a function ˆV (M t, W t, S t ). Thanks to CARA-preferences and the absence of wealth effects, the exact values of W t and S t become irrelevant for the principal s problem, and only the difference Y t = W t S t matters. Thus, we are left with the two state variables (M t, Y t ), and the principal s value can be written in the form ˆV (M t, W t, S t ) = V (M t, Y t ) State constraints Promised payments to the manager must be fully collateralized. Put differently, any uncollateralized promise Y t > M t is an empty promise. Sufficiently negative cash-flow shocks (e.g., dx t = M t ) can wipe out the firm s cash-balance within a short amount of time (t, t + dt), thereby leading to Y t+dt > M t+dt =. Under these circumstances, shareholders either renege on the promise Y t+dt and default or ask the manager to fully absorb cash-flow risk through β = 1, in order avoid liquidation. In the first case, promise keeping is violated. 12 In the second case, the manager must cover consumption needs c t = rw t and operating losses, until the firm is liquid again and financial 12 That is, the evolution of W is inconsistent with (1). This is because default at time t + dt leads to an immediate jump of payments Y t+dt >, the manager expects to receive, down to zero. Equivalently, W t+dt jumps down in absence of a Poisson shock, contradiction (1). 13

14 distress is resolved. Because the manager s consumption rw t strictly exceeds the interest earned on savings, rs t, and financial distress may prevail for an arbitrarily long time-span, she accumulates excessive debt (with positive probability), which results into a violation of the no-ponzi condition. We conclude that the only way for promise-keeping and No-Ponzi condition to hold is to liquidate as soon as Y t = M t. Thus, the principal s optimization is subject to the following state constraint: (Y, M) B = {(Y, M) : κm Y M}. (14) The general HJB-equation We can now write the principal s optimization as (r + δ) V (Y, M) = max β λ,dw,ddiv LY V (Y, M) + L M V (Y, M) + L C, L M V (Y, M) (15) subject to the state-constraint (Y, M) B. Here, L Z is the linear generator of an arbitrary stochastic process Z, and, is the quadratic variation. Note that to respect the state-constraint (Y, M) B, the principal has to engage in certain strategies when hitting the boundaries of B to keep the (Y, M) from exiting B. First, at Y = M the principal has to pays out all cash to the agent and subsequently liquidates the firm to ensure promise keeping. Second, at Y = κm, the principal has to pick wage and dividend payments in such a way as to not have Y drop below κm as response to shocks or drift exposure of Y and M. For this discussion, briefly ignore Assumption 1. Panel A in Figure 1 gives a graphical representation of the problem of controlling the process to stay in B. Consider point A strictly inside B. The principal has two strategies at his disposal: (1) Paying a dividend ddiv >, which shifts A straight left, and (2) paying a wage dw which shifts A along the 45-degree line, for example to point A, as the continuation value Y shifts 1-for-1 with the current wage payment. Let us now discuss two natural restrictions one would consider imposing on the control problem: Consider restricting the agent savings to be non-negative, i.e., S. This, destroys the first reduction in the state space, as S now has to be separably tracked. In other words, the problem with (S, W, M) = (, W, M ) is now different from the problem (S, W, M) = (Z, W + Z, M ) for any Z >. Consequently, the principal now faces a true 3-D optimization in the (S, W, M) with an additional state-constraint. Consider restricting wages to be non-negative, i.e., dw, to keep the first dimensionality 14

15 Y Panel A φ Panel B M = Y A A φ = 1 B B B O A M = κy M B B B O A φ = κ C Figure 1: Schematic Representation of the state- and strategy space reduction intact. This requires dividend payments after any shocks push (M, Y ) below the Y = κm ray. This can be seen in Panel A in Figure 1 as moving from point B to point B after a negative shift pushes O below the Y = κm ray to B, only dividend payments are effective in returning (Y, M) to within the wedge B. Such a dividend payout magnifies cash outflows, amplifying the specter of liquidity-based default. The firm will therefore want to consider building up a cash-buffer to stay away from the Y = κm ray. Consequently, the optimization is taking place on the full 2-D space (M, Y ) with a non-standard, as nonperpendicular, reflection at Y = κm. Thus, either of these restrictions leads to a relatively intractable problem requiring a numerical solution. We will next show how Assumption 1 makes the problem tractable while maintaining the key economic mechanism between liquidity and agency that we are after Second reduction of the State Space Let us return to Panel A in Figure 1, and let us consider a shift of point O to below Y = κm brought about by a negative cash-flow shock, to say point B. Recall from the discussion of non-negative wages that a dividend payout at such a point, to point B say, magnifies the cash-outflows, leading to a heavy reliance on precautionary buffer cash and a full-fledged 2-D problem. Consider instead paying a negative wage dw <, i.e., requiring the agent to contribute a small amount of her own cash to the firm, in return for a higher Y. Importantly, this payment does not violate Assumption 1, as the shock is driven by a continuous process, a Brownian motion. In essence, we are shifting 15

16 the problem up along the 45-degree line to satisfy the state-constraint (Y, M) B, here to point B. Importantly, this strategy does not change the firm s net-cash position C = M Y which measures the firms distance from default. Such negative wage contributions of course are not free, in that they entail higher deferred earnings, which in turn will require a larger risk-premium, i.e., a higher drift of Y. Thus, we are replacing a hard constraint on cash with a weaker constraint that essentially implies an increasing cost of cash the more the agent is made to contribute. To reduce the state-space further, we (1) rotate the state-space and (2) relax the problem: 1. Rotate the state space from (Y, M) to (C, ϕ), so 1 ϕ = Y/M κ and C = M Y are now representing B. In Figure 1, this rotation is represented by Panel B. Again, consider point A, which has a slack constraint Y κm. In the (C, ϕ) space, we see that paying out wages simply results in a vertical shift down, to point A. Similarly, consider point B. A dividend payment would shift the point in a north-west direction, to say point B, whereas a negative wage payment simply results in a vertical shift up to say point B. 2. Relax Assumption 1 by allowing lumpy wage payments of any sign. With unconstrained wages, ϕ can be freely adjusted up and down without affecting the firms default outlook C as discussed in point 1. Consequently, ϕ has now become a control. Assumption 1 is satisfied in the relaxed specification if the optimal control ϕ(c) is continuous (absent refinancing). Thus, we recast the principal s problem as a maximization over controls β and ϕ with the one-dimensional state C. Throughout the remainder of the paper, we refer to C also as net cash or liquidity (reserves). Utilizing (1), (2), db t =, c t = rw t, and Y = implementable contract we have ϕ 1 ϕc, for any IC and dc t = rc t dt ρr ( ) 2 (β tσ) 2 ϕt dt δa C 1 ϕ t dt + µdt + (1 β t )σdz t ddiv t C t dn t. (16) t Note that so far dw has not been explicitly specified it will be defined as the residual that implements the optimal choice of ϕ Optimization and the HJB-equation To derive the HJB, let us write the value function v(c) = v(m (W S)) = V (M, W S) = ˆV (M, W, S). Next, we conjecture that dividend payouts only occur at an upper boundary C. On 16

17 the conjectured continuation region C (, C), we have the following HJB: { ( (r + δ)v(c) = max v (C) rc ρr ( ) ) ϕc β λ,1 ϕ κ 2 (βσ)2 δa + µ + σ2 (1 β) 2 } v (C). (17) 1 ϕ 2 ( ) First, maximizing w.r.t. ϕ, as v (C) > in equilibrium, and A ϕc 1 ϕ ϕ >, we optimally set ϕ(c) = κ, (18) i.e., we pick the minimum level of cash that implements no-stealing from cash stock. With ϕ continuous, the solution to the relaxed problem is indeed the solution to the full problem. Second, maximizing w.r.t. β, the first-order conditions and the IC constraint imply that β(c) = max{λ, β (C)} with β (C) := v (C) ρrv (C) v < 1. (19) (C) Raising incentives β transfers risk to the agent and reduces the volatility of C, thereby lowering the likelihood of liquidation. Consequently, it can be optimal to provide more incentives β than required by incentive compatibility when C is low, as discussed in more detail in the next subsection. Note that the solutions to ϕ and β imply that the firm never experiences agency-based default, i.e., default triggered by C = with M = Y >. Boundary Conditions. The standard boundary conditions 13 of value-matching at default C = and smooth-pasting at the dividend payout boundary C = C are given by v() = L and v (C) = 1. (2) Recall that shareholders are not able to fully commit to their promises, and may decide to trigger liquidation if it is beneficial to them. Liquidating yields a cash payout due to the Nashbargaining assumption of θm = θ 1 κc in addition to the liquidation value L to the principal, while paying (1 θ)m = 1 θ 1 κc to the agent. Thus, for any feasible contract, we must have14 v(c) θ C + L. (21) 1 κ 13 Observe that a positive unit cash-flow shock to M at C = C leads to an increase in C of (1 β), and unit payouts of (1 β) as dividends and β as wages. Re-norming, a unit shock to C then leads to a unit dividend payout. 14 Strictly speaking, we must have v(c) θ 1 κ C + L for all C [, C], but from v (C) 1 v (C) it is sufficient to check this condition at C = C. 17

18 If constraint (21) is slack, the payout boundary satisfies the optimality or super-contact condition v (C) = (22) In other words, if payouts are optimally made at C = C with (21) slack, then the shareholders effective risk-aversion vanishes at C. Thus, whenever (21) holds with equality and v (C) <, the shareholders limited commitment combined with moral hazard κ constrain the firm in optimally managing liquidity risks. Note that constraint (21) is always slack if θ 1 κ promise keeping, as (1 θ)m < Y 1 < θ 1 κ.15 For at the candidate payout boundary C defined by (22). Proposition 2. Let C solve (5). Then, the following holds true: < 1, which is the case when a liquidation would not violate θ 1 κ > 1, we simply check condition (21) i) The shareholders value function V ( ) satisfies V ( ) = v(c), where the function v( ) is twice ii) continuously differentiable, i.e., v C 2. The principal s payoff is given by a function v, that solves the HJB-equation (17) subject to v() L = v (C) 1 = and either v (C) = or v(c) = θc/(1 κ) + L. iii) The value function v is strictly concave [, C) with v (C) >. 4 Analysis Unless specified otherwise, we assume that parameters are such that the payout boundary is optimally determined by the super-contact condition, i.e., v (C) = Performance-Pay & Hedging Through Labour Contracts In this section, we analyze the pay-performance sensitivity β. For clarity of exposition, let us for the time being assume that λ = θ =, so that β = β. The assumption λ = is equivalent to the absence of the agency problem in terms of stealing out of cash-flow, but does not preclude stealing from cash-stock, i.e., κ >. Absent liquidity concerns, it is optimal for the principal not to expose the risk-averse manager to any cash-flow shocks, i.e., to set β = λ =. However, in the presence of liquidity concerns, 15 This is because v( C) L = v( C) v() > C, as v (C) 1 with the inequality being strict for some C. Hence, the super contact condition holds if θ C Cθ κ As mentioned in the preceding footnote, a sufficient condition for this is θ < 1 κ. 18

19 Figure 2: Benchmark value function: The parameters are µ =.25, r =.1, δ =.25, λ = κ =.4, σ =.75, θ = and ρ = 7. shareholders become increasingly risk-averse as cash-reserves dwindle and would optimally like to hedge liquidity risk through labour contracts by setting incentive pay β >. Incentive pay transfers risk to the agent, in that the volatility of the liquidity reserves, dc/dx = σ(1 β), decreases in β for β < 1. Consider the benefit of increasing β: v(c) β v (C)ρrβσ 2 }{{} + (1 β)σ 2 ( v (C)) }{{}. Risk-Compensation;< Reduction in Cash-Flow volatility;> Increasing β makes C less volatile and reduces the likelihood that the firm runs out of cash but also requires a risk-compensation to the agent, as her wage has become more volatile. When the firm has low cash holdings, a reduction in volatility is particularly beneficial, since v (C) is large. On the other hand, the marginal value of cash of the firm v (C) is pronounced under distress, so that the drift of promised wages required as risk-compensation is also very costly. Intuitively, the optimal β implements a risk-sharing solution that balances the agent s constant absolute risk-aversion ρ against the shareholders state-dependent absolute risk-aversion v (C)/v (C). The firm hedges more strongly through labour contracts for low net-cash positions, i.e., β (C) > for C >, whereas it absorbs all risk at the payout boundary, β (C) =. That is, compensation becomes more equity like when the firm undergoes financial distress and has little cash. In practice, firms with little cash often are start-ups and young firms, where it is indeed well documented that their employees are often rewarded with stock. When λ >, the firm s risk-sharing is constrained by β λ. Thus, risk-sharing is constrained for high levels of C in that due to IC constraint the principal can never fully insure the agent, even 19

20 at the payout boundary as β () = λ. Furthermore, the incidence of negative wages dw < rises for low C, in that = dϕ = d(y/m) implies dw = µ w dt + β(c) κ 1 κ dz. In other words, the model predicts an increased propensity of managers to pledge private assets in response to negative cash-flow shocks for low liquidity firms, something that is common in both start-ups and firms in financial distress. In the special case of λ = κ, we have β(c) κ 1 κ. Therefore, negative wages in response to cash-flow shocks occur exactly when the risk-sharing considerations outweigh the agency issues. We summarize our findings in the following corollary. Corollary 1. Let C solve (5). Then, the following holds true: i) There exists C [, C), so that the pay-performance sensitivity β (weakly) decreases in on [C, C]. In particular, β (C) C < on [C, C]. If σ is sufficiently low, then C = ii) There exists a unique value Ĉ [, C], such that β(c) > λ for C < Ĉ. If λ is sufficiently small, it follows that Ĉ >. iii) The loading of wages on the cash-flow shocks is given by β(c) κ 1 κ more prevalent for low-cash firms. and thus negative wages are 4.2 Risk-sharing vs retained earnings as liquidity management tools In our setting, the firm has two distinct but connected tools to manage liquidity risks: The firm can hedge liquidity risk through labour contracts and provide particularly highpowered incentives β during financial distress when C is close to zero. The firm can increase retained earnings accumulation, as proxied by the dividend boundary C. All else equal, a higher payout boundary C implies higher average net-cash holdings. Let us first establish the following analytic results regarding comparative statics: Corollary 2 (Hedging through high powered incentives). For a firm under distress, i.e., C, β(c), the analytic comparative statics are summarized in the first row of Table 1. Corollary 3 (Hedging through cash reserves). For the target cash-holdings C, the analytic comparative statics are summarized in the second row of Table 1. 2

21 β (C ) κ (κ sufficiently large) σ + (ρ, λ sufficiently low), otherwise µ + ρ (low ρ), (otherwise) (low ρ), + (high ρ) λ (low λ), + (otherwise) (low & high λ) δ (κ sufficiently large) θ (high θ, κ), (otherwise) (high θ, κ), (otherwise) C Table 1: Comparative statics. Next, we will show numerically that these two liquidity management tools are substitutes by analyzing the following experiments: consider constraining the principal to a sub-optimal strategy in one of the two liquidity management tools (i) an exogenously too high β(c), or (ii) an exogenously too low C. From our previous discussions, a situation in which the IC constraint (19) is binding is essentially experiment (i) and can thus be proxied for by comparative statics w.r.t. λ, whereas a situation in which the commitment constraint (21) is binding is essentially experiment (ii) and can thus be proxied for by comparative statics w.r.t. θ. In our discussion below, avg β refers to the equal-weighted integral C β(c)dc/c. Changing λ = κ. Let us consider varying the degree of agency friction as measured by the stealing efficiency λ = κ. Column 1 of Figure 3 shows the behaviour of C and avg β (solid black lines) when varying λ = κ. The avg β increases mechanically as we are raising the floor on β(c) (dashed red line) via the IC constraint. In response to this increased risk-sharing through labor contracts, the need for retained earnings decreases and C optimally shrinks. Moreover, more severe moral hazard reduces firm value and thereby also overall hedging demand. Not shown here is that numerically there is almost no movement in β(). Changing θ. Let us consider varying the degree of commitment by the manager as measured by the bargaining weight θ. As long as (21) is slack changes in θ have no impact on any of the principal s choices. However, once θ is high enough and (21) starts binding the firm has to use an inefficiently low payout boundary C. Column 2 in Figure 3 illustrates. Constraint (21) starts binding at θ.85, and any further increase in θ reduces the payout-boundary C. To counteract this deterioration in liquidity management via retained earnings, the principal increases hedging through labor contracts by increasing the pay-performance sensitivity of wages, as indicated by an 21

22 Figure 3: Comparative statics w.r.t. λ = κ (Column 1), w.r.t. θ (Column 2), w.r.t. σ (Column 3), top row C, bottom row σ-scaled avg β. The solid black lines depict the object described on the y-axis, the dashed red line depicts the IC constraint (19), the thin vertical dashed red line depicts the parameter value in our benchmark. 22

23 increase in avg β. Changing σ. Let us discuss changing the dynamics of the cash-flow generating process. Here, the effects are more complex in that some non-monotonicity appears. First, consider an increase in σ. A higher σ in a pure risk-sharing model, that is with λ =, will lead to a higher payout boundary C as default has now become more likely, holding everything else constant. Non-monotonicity can only arise when the commitment constraint (21) starts binding and then follows closely the explanations in the discussion regarding θ. Column 1 in Figure 8 shows the situation in which λ >. We see that C is non-monotone even in the absence of (21) binding. The intuition is as follows: higher σ raises the risk of liquidation and requires more intense risk-management, so that C and avg β increase. However, due to agency conflicts, the agent must be provided costly incentives β λ, even if this is not optimal from a pure risk-management perspective. Consequently, severe agency conflicts drain the firm value and reduce the overall hedging demand. The latter effect dominates, when σ and λ are sufficiently large and the agent requires a high risk-premium in response to performance-pay. Changing ρ, δ and µ. The comparative statics of ρ, δ and µ are relegated to appendix E. Since δ essentially captures carry-cost of cash, C not surprisingly decreases in δ. Moreover, when the agent is more risk-averse, incentive-pay and therefore hedging through labour contracts becomes more costly, so that the firm hedges more through retained earnings instead, in that C increases in ρ. On the other hand, moral hazard has more bite for larger ρ, which in turn implies that overall firm value decreases in ρ. As a result, liquidation gets less inefficient, which calls for less hedging of liquidity risks. This leads to non-monotonic comparative statics of C wrt. ρ. 4.3 Stock Return Volatility and Agency Conflicts In this section, we discuss how firm agency conflicts impact the firm s stock returns: dr t = ddiv t + dv(c t ) v(c t ) = r + δ + ddiv t v(c t ) + Σ tdz t. (23) Of particular interest is the stock-return volatility Σ t = Σ(C t ) where Σ(C) = σ(1 β(c)) v (C) v(c). (24) 23